optimize and trade-off between QEC cycle parameters, assess the merits of feedback control, predict gains from future improve- ments in physical qubit performance, and quantify decoder performance. We compare an algorithmic decoder using minimum-weight perfect matching (MWPM) with homemade weight calculation to a simple look-up table (LT) decoder, and weigh both against an upper bound (UB) for decoder performance obtainable from the **density**-**matrix** simulation. Finally, we make a low-order approximation to extend our predictions to the distance-5 Surface-49. The combination of results for Surface-17 and Surface-49 allows us to make statements about code scaling and to predict the code size and physical qubit performance required to achieve break-even points for memory and computa- tional performance.

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In the Wigner model, due to the spontaneous emission, ⁄ ́ is proportional to as expressed by the second term on the right-hand side of Eq. (9). In this case, the probability of occupying the final state increases by spontaneous emission from electrons occupying the initial state but decreases by spontaneous emission from electrons occupying the final state. Then, the Wigner model is more exact than the **density** **matrix** because the spontaneous emission from the final state is also taken into account. The drawback associated to the trace preserving property assumed in the **density** **matrix** treatment becomes pronounced as the rate of the spontaneous emission increases. As will be shown in the next section, in the practical regime of QFEL operation ( ), the **density** **matrix** model is quite applicable where a very good agreement with the discrete Wigner model is observed. It is also noticed that, by comparing Eq. (10) and Eq. (29), the lifetime of coherence in the Wigner model is ⁄ , twice as large as that appeared in the **density** **matrix** model. In fact, this does not cause a significant difference between both models since the spontaneous emission rate ( ̅ ⁄ ) in the

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Terahertz (THz) quantum cascade lasers (QCL) are currently increasing in popularity. It is expected to become the main source of emerging terahertz radiation technology and applications. However to produce the device within the application specification is costly and time consuming. This is because the manufacturing process of the superlattice growth and the device processing and testing are long and expensive processes. Thus a prediction tool is needed to overcome the problems in designing and producing THz QCL within the needed optical expectation. The **density** **matrix** method is used to calculate the performance of this device electronically and optically. The result obtained was compared to the experimental result conducted by previous researchers. The calculation result showed that the gain is 20 cm −1 when the population inversion occurs at threshold current

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The solution is significantly easier in the case when the properties of the system under analysis make it possi- ble to predict what representation is used to bring the **density** **matrix** to its diagonal pattern. As applies to this case, it is time to solve the Equation (4.3). Any solutions sourced from the above equation can exhibit certain interesting features related to its nonlinearity and particular dependence of kernel ε nn ′ on quantum numbers n

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My PhD work is focused on the approximation methods of the **density** **matrix**. The basis of **density** **matrix** methods lies in the commutativity between the **density** **matrix** and the Kohn- Sham Hamiltonian **matrix**, i.e., the **density** **matrix** can be written as a **matrix** function of the Kohn-Sham Hamiltonian. The rise of linear-scaling **density** functional theory methods led to the applications **matrix** function approximations to **density** functional theory. A good reference that describes the various ways to approximate a **matrix** function is the book by Higham [32]. To my knowledge, there has been at least one paper published in the linear-scaling **density** functional theory literature using the approaches discussed by Higham: from polynomial based approximations using spectral Gauss quadratures to rational function approximations. However, I think there is still room in adapting the implementation of existing approximiations to better suit the architecture of newest supercomputers.

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Currently, no experimental or theoretical data on coherent phenomena in QCLs in a magnetic field are available. Since the energy spectra in such structures is discrete, it is reason- able to expect that coherent effects are more significant than for QCLs without magnetic field. The aim of this work is to present a quantum-mechanical theory of transport and gain properties of QCLs in an external magnetic field. For that purpose, we derived quantum kinetics equations for QC structures in a magnetic field, based on the **density** **matrix** formalism, which include interaction of electrons with LO phonons and optical field. Furthermore, we obtained the cor- responding equations in the Markovian approximation, from which the semiclassical Boltzmann transport equations can be recovered. A comprehensive analysis is performed for an example GaAs/ Al 0.3 Ga 0.7 As QCL and nonequilibrium steady

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Combined with J -engine techniques [33, 34, 35] RI gives tremendous speed-ups [8, 9] for Coulomb-like contribu- tions. Although still applicable to exchange [10, 11, 12, 13, 16, 17, 20, 21], the RI methodology does not exhibit the same favorable performance gains as for the Coulomb integrals. Alternative schemes such as the auxiliary-**density**-**matrix** method (ADMM) [36, 37] and the chain-of-spheres algorithm (COSX) [38] have therefore been developed specifically for the exchange contribution. In ADMM, the exchange energy is split into two parts. One part consists of the exact HF exchange evaluated in a small auxiliary atomic basis set (from an auxiliary **density** **matrix**); the second part is a first-order correction term, evaluated as the difference between the generalized gradient approximation (GGA) exchange in the full and auxiliary basis sets. The auxiliary **density** **matrix** can be obtained by means of projection from the full **density** **matrix** fulfilling various constraints, as discussed by Guidon et al. [36] and Merlot et al. [37]. The COSX approxima- tion builds on the use of semi-numerical integration techniques, first introduced by Friesner in the pseudo-spectral method [39, 40, 41] and later refined in the COSX approach by Neese et al. [38]. In this approach the Coulomb potentials of products of two one-electron basis functions are evaluated analytically on a grid, followed by a numerical integration over the second electron. Reported speed-ups are of up to two orders of magnitude relative to calculations involving explicit exchange-**matrix** formation [38]. In this work we explore how these techniques may be exploited further in the calculation of molecular properties using response theory.

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in which the block subscripts describe interaction terms be- tween the central (C), upper (U), and lower (D) periods. The off-diagonal elements correspond to the coupling between the periods labeled by the corresponding indices. Every block in (1) has the size of N × N , where N is the number of states in a sin- gle module. We employ wave functions from a single period as the basis for the **density** **matrix** operator, and the corresponding **density** **matrix** will have similar form to the Hamiltonian in (1). This would generally result in nine block equations; however, we can greatly simplify the problem if we neglect direct cou- pling between the “U” and “D” modules, H UD = H DU = 0, and

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In this work, we provided a comprehensive discussion of the properties of the one-particle **density** **matrix** in one- dimensional systems of interacting fermions in the pre- sence of disorder. We showed that the eigenstates of the OPDM, which form complete sets of single-particle sta- tes, computed in individual many-body eigenstates, are delocalized in the ergodic phase and localized in the MBL phase. The eigenvalues, after a suitable reordering, un- veil the Fock-space structure of MBL many-body eigen- states: they are weakly dressed Slater determinants [1] since most eigenvalues are close to either one or zero and the occupation spectrum has a discontinuity at an emer- gent Fermi edge in the MBL phase. This suggests a close analogy of the MBL phase to a zero-temperature Fermi liquid and is consistent with the existence of localized quasiparticles in the MBL phase. We relate these findings to the local integrals of motion introduced to describe the

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More over, results of the experiments depend on tun- ing of the coincidence events, and this tuning can be arranged in such a way that one can get not only viola- tion, but even super violation of the Bell’s inequalities, as is shown in [13]. There are a lot of references, which show diﬀerent defects of the conducted experiments, and at present there is an opinion that still no experiment was made without a loophole. I suppose that proposed here the feasible experiment proving contradiction of the **density** **matrix** will stop the useless search for loophole free experiment to prove entanglement of far separated particles.

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As mentioned earlier, DEAP Database has been used in our proposed automatic emotion recognition system. Such dataset consists of more than one-thousand signals recorded from 32 participants who have been asked to watch 40 videos containing the three emotional cases for each dimension. The two-dimensional phase space trajectory of the signal has been reconstructed using the delay time embedding method. Then the phase space plots of the signals have been divided by a grid of 20x20 squares and the numbers of points C(i,j) within each square have been computed to form the phase space **density** **matrix** C. Four features have been extracted from the GLCM **matrix** from both Beta and Gamma band of the six EEG channels (Fp1, Fp2, Cz, Fz, F3, and F4) to form the features vectors. There are several approaches to assess the significance of the extracted features such as Fisher projection, ANOVA, and sequential feature selection. In this paper, one-way analysis of variance (ANOVA) [19] has been chosen to test the significances of the extracted features. Error! Reference source not found. shows the p-value for all features that have been obtained from applying ANOVA test.

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Here we included a more detalied account on the results presented in the paper. We present statistics, mean errors, standard deviations and maximum absolute errors, compared to aug-pcseg- 4 reference calculations for three types of calculations: full, df-J and admm. Here full referes to a calculation combining J -engine for Coulomb and LinK for exchange (i.e. without any approxima- tion), df-J to the combination of **density**-fitting for Coulomb and LinK for exchange, and admm to the combination of **density**-fitting for Coulomb and admm for exchange.

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Many interesting physical systems have a complex action which impedes Monte Carlo simulations due to the notorious sign problem. In this category belongs QCD at finite baryon **density**, Yang- Mills theories in the presence of a θ-term, systems of strongly correlated electrons and many other interesting physical systems. The Complex Langevin (CL) algorithm has been advocated as one of the most promising routes for a solution of the sign problem [1, 2]. Despite some successes in toy models, the algorithm seems to have pathological issues such as converging to the wrong theory. Detailed studies have also revealed that the criteria which were put forward in order to guarantee a correct result are not fulfilled in practice in many cases of interest such as cold and dense QCD close to the chiral limit. An update of the current status can be found in [3]. In this talk we discuss a Random **Matrix** Theory (RMT) model of QCD at nonzero baryon **density**, which has an exponentially hard sign problem, but serves as an excellent test bed for new algorithms since it can be solved analytically. This model is based on a model for QCD at nonzero nonzero temperature or imaginary chemical potential [4] and was extended to QCD at nonzero chemical potential by Stephanov in [5].

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be found using our quantum relative entropy with a suitable uniform prior density matrix.. 27.[r]

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However, the excessive growth of biomass on the me- dia can lead to washout of bioparticles (particles covered by biomass) from a reactor since the biomass loading can increase to such an extent that the bioparticles began to be carried over from the reactor [9,10]. The application of the inverse FBBR, in which a bed consisting of low **density** (**matrix** particle **density** smaller than that of liquid) particles expands downwards during fluidization, allows the control of biomass loading and provides the high oxygen concentration in the reacting liquid media [1,3].

are conserved for all Hamiltonians, but these cannot con- tribute to the stationary state in the most general case be- cause otherwise (among the many strange consequences) no system will ever thermalise. Furthermore, taking all these projector in to account would be equivalent to time- average the complicated many-body dynamics, but this will result in a trivial prediction that has no connection with the economy of an ensemble description of statistical physics. Indeed such an ensemble would retain all of the information about the initial state, rather than information about only a minimal set of integrals of motion. To clarify this point, it has been understood recently [16, 19, 25] that only local integrals of motion should be used in Eq. (6) as long as we are interested in the expectation values of local observables such as the reduced **density** **matrix** (an integral of motion is said to be local if it can be written as an integral–sum in the case of a lattice model– of a given local current, in the spirit of Neither theorem). Thus, from now on, we will always refer to the GGE in Eq. (6) in which only local integrals of motion are included.

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The electric quadrupole moments for some scandium isotopes ( 41, 43, 44, 45, 46, 47 Sc) have been calculated using the shell model in the proton-neutron formalism. Excitations out of major shell model space were taken into account through a microscopic theory which is called core polarization effectives. The set of effective charges adopted in the theoretical calculations emerging about the core polarization effect. NushellX@MSU code was used to calculate one body **density** **matrix** (OBDM). The simple harmonic oscillator potential has been used to generate the single particle **matrix** elements. Our theoretical calculations for the quadrupole moments used the two types of effective interactions to obtain the best interaction compared with the experimental data. The theoretical results of the quadrupole moments for some scandium isotopes performed with FPD6 interaction and Bohr-Mottelson effective charge agree with experimental values.

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References [36, 41] show that the OM basis provides information about the physical state of a system on the example of the Holstein polaron. We continue along those lines and study the exact structure of the OM basis states in the Bose-Bose resonance model – a model describing two bosonic particle species interacting via a Feshbach interaction [42,43]. For our choice of parameters, this model features a rich phase diagram including a Mott insulating phase, a molecular quasi-condensate phase and a phase where atoms and molecules quasi condense [43, 44]. For the three limiting cases deep inside those phases we approximate the local reduced **density** **matrix** using perturbation theory and the exact expression for a Bose- Hubbard model without interaction and thereby get an estimate for the optimal mode states and their weights which perfectly matches the numerically exact data obtained using the DMRG method. We find that the optimal mode states are different deep in the three phases of this specific model and we therefore conclude that they can be used to distinguish between states deep in the respective phases. Also, we use the DMRG method to calculate the weight spectrum and, derived from the weight spectrum, the local von Neumann entropy as well as the OMs as we cross two phase boundaries. This complements earlier works [45, 46] where the local von Neumann entropy was used to detect phase transitions in spin- and fermionic models. Reference [46] showed that the local von Neumann entropy works as an indicator of phase transitions in some but not all cases and that the von Neumann entropy calculated from two sites is a better indicator. In this thesis we follow up with the study of the local reduced **density** **matrix** of a single site. Because we treat a model with bosonic degrees of freedom, the reduced **density** **matrix** has a much richer structure. We find that, for our model, the local von Neumann entropy (and thus the weight spectrum) as well as the OM states show features in the vicinity of the phase boundaries.

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Nuclear Magnetic Resonance has a theory which is well established and very well validated. The aim of this lec- ture was to provide a fast introduction to this theory, more detailed and in-depth descriptions can be found in several reference books [1,2,5,4,3,6,7]. The presentation is orga- nized in such a way. First, after introducing the nuclear spin, elements are provided explaining why quantum me- chanics and statistical physics are needed to properly de- scribe the evolution of the magnetization and thus predict the result of an NMR experiment. In a second step, the **density** **matrix** operator which allows such a treatment is defined and the main theorems allowing the calculation of its value at thermal equilibrium, its time-domain evolution, its expression in a rotating frame and its reduction to the spin system are given. Two simple examples of calcula- tions (free evolution and rf excitation) performed in this framework are given. The next section is devoted to the classical approach based on the Bloch equations. The ques- tion of sensitivity in NMR is stressed and the equivalence between this approach and the **density** **matrix** one with its limits is shown. Then, the different main interactions present in conventional NMR (chemical shift, dipolar and quadrupolar interactions, scalar coupling) are described. The aspects of truncated Hamiltonian and interaction sym- metries are considered. The last section is devoted to a fast introduction to relaxation due to randomly time-dependent Hamiltonian. The master equation of relaxation is derived and the principal definitions (self and cross-relaxations, auto-correlation and cross-correlation contribution to re- laxation) are given. Finally the importance and relevance of spectral **density** functions are discussed.

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interaction to generate model space vectors with harmonic oscillator potential as a single particle wave function. OXBASH code is used to carry this calculations and the program of Core, Valence, Tassie (CVT) written in FORTRAN go language to calculate the Electric quadrapole moments between excited states themselves. All of these calculations have been carried through model space vectors only. One body **density** **matrix** elements (OBDM) for ground and Excited states is calculated in order to carry the calculations using single particle Transition **matrix** elements between excited states theme selves.

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